Maxima and Minima of Functions of Two Variables

Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points. More on Optimization Problems with Functions of Two Variables in this web site.

Theorem

Let f be a function with two variables with continuous second order partial derivatives fxx, fyy and fxy at a critical point (a,b). Let

D = fxx(a,b) fyy(a,b) - fxy2(a,b)

a) If D > 0 and fxx(a,b) > 0, then f has a relative minimum at (a,b).
b) If D > 0 and fxx(a,b) < 0, then f has a relative maximum at (a,b).
c) If D < 0, then f has a saddle point at (a,b).
d) If D = 0, then no conclusion can be drawn.

Examples with Detailed Solutions

We now present several examples with detailed solutions on how to locate relative minima, maxima and saddle points of functions of two variables. When too many critical points are found, the use of a table is very convenient.

Example 1

Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by

f(x , y) = 2x2 + 2xy + 2y2 - 6x

.
Solution to Example 1:Find the first partial derivatives fx and fy.
fx(x,y) = 4x + 2y - 6
fy(x,y) = 2x + 4y
The critical points satisfy the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence.
4x + 2y - 6 = 0
2x + 4y = 0
The above system of equations has one solution at the point (2,-1).
We now need to find the second order partial derivatives fxx(x,y), fyy(x,y) and fxy(x,y).
fxx(x,y) = 4
fyy(x,y) = 4
fxy(x,y) = 2
We now need to find D defined above.
D = fxx(2,-1) fyy(2,-1) - fxy2(2,-1) = ( 4 )( 4 ) - 22 = 12
Since D is positive and fxx(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1).
The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6).

Example 2

Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by